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We have proved the following statement, but wonder if this result is actually known (reference??)

It solves the following problem. Suppose you have a (possibly) hollow chocolate egg whose outer boundary is the boundary of a convex body. If all the chocolate is within distance $0<\delta<1$ of the outer boundary of the egg, what is the maximum quantity of chocolate that can possibly be contained within a unit ball?

Let $B(x)$ denote the ball with the unit radius centred at $x$; we also denote by $N_\delta(X)$ the $\delta$-neighbourhood of $X$, i.e. the set of points whose Euclidean distance to $X$ is less than $\delta$.

Lemma. Let $0<\delta<1$ and let $\mathcal{B}$ be a convex body in $\mathbb{R}^d$. For every $x \in \mathbb{R}^d$, we have \begin{equation} \lambda\left(B(x) \cap \mathcal{B} \cap N_\delta(\mathcal{B}^c)\right) \le c_d(1 - (1-\delta)^d)\,, \end{equation} where $c_d=\lambda(B(x))$ is the volume of the unit ball in $\mathbb{R}^d$ and $\lambda(\cdot)$ denotes the volume of the set.

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